by Nikolai Golovchenko debugged by Daniel Schouten [daniels at XS4ALL.NL]
Well, let's calculate y=2^x! x is Q6.10 notation (always negative with implicit sign). Output y has Q0.16 notation (always less then 1). Therefore x should not be zero (2^0 = 1).
y is in the range from 0.9993 (x = -1/1024) to 5e-20 (x = -64 + 1/1024). Okay....
y=2^x y=2^-(32*x.15+16*x.14+...1*x.10 + 0.5*x.9+0.25*x.8+...) \========================/ \===================/ x_int x_frac x_int = 0:1:63 x_frac = 0:1/1024:(1-1/1024) y=2^-(x_int+x_frac) y=2^(-x_int) * 2^(-x_frac) y= (2^(-x_frac)) >> x_int =========================
So we first find 2^-x_frac using a table and then shift the result right x_int times.
First we should convert the fraction part through a table to a new fraction part. 2^(-0..-1) = 0.5..1
; y = 2^x2 ;Input: x2hi:x2lo ;Output: yhi:ylo ;Form table index from 4 higher bits of x2 fraction part rlf x2lo, f rlf x2ho, w movwf index rlf x2lo, w rlf index, f ;here middle 6 bits of x2lo contain multiplier for linear ;interpolation, and index has 4 lower bits of table pointer rlf index, w ;multiply index by 2 for 16 bit ;entries addressing andlw 0x1E ;and clear lsb and higher bits movwf index ;Read first point to temphi:templo call exp2_table ;clear result and subtract the first point from it clrf yhi clrf ylo movf templo, w subwf ylo, f movf temphi, w skpc incfsz temphi, w subwf yhi, f ;Read next point to temphi:templo incf index, f incf index, f call exp2_table ;find difference with previous point to temphi:templo movf ylo, w addwf templo, f movf yhi, w skpnc incfsz yhi, w addwf temphi, f ;multiply the difference by the next 6 fraction bits in ;x2lo and divide by 64. Then subtract it from the current ;result in yhi:ylo movlw 6 movwf index ;use index as a loop counter rrf x2lo, f ;align the 6 bits to the lsb clrf temp ;we will use another temp ;for mul purposes rrf yhi, f ;rotate y right 2 times rrf ylo, f ;to make easy multiplication. rrf temp, f ;lower bits will go to temp rrf yhi, f rrf ylo, f rrf temp, f exp2loop rrf x2lo, f ;get next multiplier bit bnc exp2loopnext ;skip subtraction if carry is clear movf templo, w subwf temp, f movf temphi, w skpc incfsz temphi, w subwf ylo, f skpc decf yhi, f exp2loopnext rrf yhi, f rrf ylo, f rrf temp, f decfsz index, f goto exp2loop ;shift the result 8 bits left movf ylo, w movwf yhi movf temp, w movwf ylo ;Now we will shift y right the number of times ;equal to the integer part of x2 clrc ;align integer bits rrf x2hi, f ;to lsb clrc rrf x2hi, f test x2hi skpnz return ;done exp2loop2 clrc rrf y2hi, f rrf y2lo, f decfsz x2hi, f goto exp2loop2 return ;done
Now we need to calculate the table...
1-2.^(0:-1/16:-1)
ans =
Columns 1 through 4 0 0.04239671930143 0.08299595679533 0.12187391981335 Columns 5 through 8 0.15910358474629 0.19475483402537 0.22889458729603 0.26158692703025 Columns 9 through 12 0.29289321881345 0.32287222653155 0.35158022267450 0.37907109396326 Columns 13 through 16 0.40539644249864 0.43060568262165 0.45474613366737 0.47786310878629 Column 17 0.50000000000000
Multiplied by 65536 and rounded
round(65536*(1-2.^(0:-1/16:-1)))
ans =
Columns 1 through 6 0 2779 5439 7987 10427 12763 Columns 7 through 12 15001 17143 19195 21160 23041 24843 Columns 13 through 17 26568 28220 29802 31317 32768 »
So, the table would be:
exp2_table movf index, w call exp2tableStart movwf templo incf index, w call exp2tableStart movwf temphi return exp2tableStart addwf PCL, f DT 0 & 0xFF, 0 >> 8, 2779 & 0xFF, 2779 >> 8 DT 5439 & 0xFF, 5439 >> 8, 7987 & 0xFF, 7987 >> 8 DT 10427 & 0xFF, 10427 >> 8, 12763 & 0xFF, 12763 >> 8 DT 15001 & 0xFF, 15001 >> 8, 17143 & 0xFF, 17143 >> 8 DT 19195 & 0xFF, 19195 >> 8, 21160 & 0xFF, 21160 >> 8 DT 23041 & 0xFF, 23041 >> 8, 24843 & 0xFF, 24843 >> 8 DT 26568 & 0xFF, 26568 >> 8, 28220 & 0xFF, 28220 >> 8 DT 29802 & 0xFF, 29802 >> 8, 31317 & 0xFF, 31317 >> 8 DT 32768 & 0xFF, 32768 >> 8 IF (($ - 1) >> 8) - ((exp2tableStart + 1) >> 8) != 0 error 'exp2 table crossed 8-bit boundary' ENDIF
And the cleaned up version is below... Yeah, this one was harder and not really the reverse of log!
Don't forget to check x and y for zero in x^y.
#include <p16f84.inc> cblock 0x20 xhi, xlo templo, temphi, temp index x2hi, x2lo endc #define tval 65300 movlw tval & 0xFF movwf xlo movlw tval >> 8 movwf xhi call log2 nop call exp2 nop ;************************************************ ; EXP2 routine for fixed point unsigned ; 16 bit values in 6.10 notation (implicit minus) ; ; Input: x2hi:x2lo unsigned Q6.10 (implicit minus) ; Output: xhi:xlo unsigned Q0.16 ; Temporary: index, templo, temphi, temp ; ; Maximum error: 0.025% (for x2 < 1) ; ; Size: 121 words ; Time: min 2+8+18*2+10+16+10*6-1+6+2=139 ; max 2+8+18*2+10+16+17*6-1+6+1+14+6*16-1+2=291 ; ; Note: Zero input is illegal! Will result in ; zero output. (2^0 = 1, but x is always less ; than 1) ; ; 30 Sep 2000 by Nikolai Golovchenko ; 02 Oct 2000 debugged by Daniel Schouten [daniels at XS4ALL.NL] ;************************************************ exp2 ;Form table index from 4 higher bits of x2 fraction part rlf x2lo, f rlf x2hi, w movwf index rlf x2lo, w rlf index, f ;here middle 6 bits of x2lo contain multiplier for linear ;interpolation, and index has 4 lower bits of table pointer rlf index, w ;multiply index by 2 for 16 bit ;entries addressing andlw 0x1E ;and clear lsb and higher bits movwf index ;Read first point to temphi:templo call exp2_table ;clear result and subtract the first point from it clrf xhi clrf xlo movf templo, w subwf xlo, f movf temphi, w skpc incfsz temphi, w subwf xhi, f ;Read next point to temphi:templo incf index, f incf index, f call exp2_table ;find difference with previous point to temphi:templo movf xlo, w addwf templo, f movf xhi, w skpnc incfsz xhi, w addwf temphi, f ;multiply the difference by the next 6 fraction bits in ;x2lo and divide by 64. Then subtract it from the current ;result in xhi:xlo movlw 6 movwf index ;use index as a loop counter rrf x2lo, f ;align the 6 bits to the lsb clrf temp ;we will use another temp ;for mul purposes rrf xhi, f ;rotate x right 2 times rrf xlo, f ;to make easy multiplication. rrf temp, f ;lower bits will go to temp rrf xhi, f rrf xlo, f rrf temp, f exp2loop rrf x2lo, f ;get next multiplier bit bnc exp2loopnext ;skip subtraction if carry is clear movf templo, w subwf temp, f movf temphi, w skpc incfsz temphi, w subwf xlo, f skpc decf xhi, f exp2loopnext rrf xhi, f rrf xlo, f rrf temp, f decfsz index, f goto exp2loop ;shift the result 8 bits left movf xlo, w movwf xhi movf temp, w movwf xlo ;Now we will shift x right the number of times ;equal to the integer part of x2 clrc ;align integer bits rrf x2hi, f ;to lsb clrc rrf x2hi, f tstf x2hi skpnz return ;done movlw 17 ;limit shifts number to 16 maximum subwf x2hi, w movlw 16 skpnc movwf x2hi ;if fractional part is zero than set carry, else ;clear carry movf xlo, w iorwf xhi, w setc skpz clrc ;limit number of si ;perform shifts exp2loop2 rrf xhi, f rrf xlo, f clrc decfsz x2hi, f goto exp2loop2 return ;done exp2_table movf index, w call exp2tableStart movwf templo incf index, w call exp2tableStart movwf temphi return exp2tableStart addwf PCL, f DT 0 & 0xFF, 0 >> 8, 2779 & 0xFF, 2779 >> 8 DT 5439 & 0xFF, 5439 >> 8, 7987 & 0xFF, 7987 >> 8 DT 10427 & 0xFF, 10427 >> 8, 12763 & 0xFF, 12763 >> 8 DT 15001 & 0xFF, 15001 >> 8, 17143 & 0xFF, 17143 >> 8 DT 19195 & 0xFF, 19195 >> 8, 21160 & 0xFF, 21160 >> 8 DT 23041 & 0xFF, 23041 >> 8, 24843 & 0xFF, 24843 >> 8 DT 26568 & 0xFF, 26568 >> 8, 28220 & 0xFF, 28220 >> 8 DT 29802 & 0xFF, 29802 >> 8, 31317 & 0xFF, 31317 >> 8 DT 32768 & 0xFF, 32768 >> 8 IF (($ - 1) >> 8) - ((exp2tableStart + 1) >> 8) != 0 error 'exp2 table crossed 8-bit boundary' ENDIF ;************************************************
file: /Techref/microchip/math/power/2x-16b-ng.htm, 10KB, , updated: 2000/10/3 14:06, local time: 2024/8/21 20:17,
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